On double domination in graphs
Jochen Harant ; Michael A. Henning
Discussiones Mathematicae Graph Theory, Tome 25 (2005), p. 29-34 / Harvested from The Polish Digital Mathematics Library
Access to full text
Full (PDF)

In a graph G, a vertex dominates itself and its neighbors. A subset S ⊆ V(G) is a double dominating set of G if S dominates every vertex of G at least twice. The minimum cardinality of a double dominating set of G is the double domination number γ×2(G). A function f(p) is defined, and it is shown that γ×2(G)=minf(p), where the minimum is taken over the n-dimensional cube Cⁿ=p=(p₁,...,pₙ)|pi∈IR,0≤pi≤1,i=1,...,n. Using this result, it is then shown that if G has order n with minimum degree δ and average degree d, then γ×2(G)≤((ln(1+d)+lnδ+1)/δ)n.

Publié le : 2005-01-01
EUDML-ID : urn:eudml:doc:270158 
@article{bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1256,
     author = {Jochen Harant and Michael A. Henning},
     title = {On double domination in graphs},
     journal = {Discussiones Mathematicae Graph Theory},
     volume = {25},
     year = {2005},
     pages = {29-34},
     zbl = {1073.05049},
     language = {en},
     url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1256}
}

Jochen Harant; Michael A. Henning. On double domination in graphs. Discussiones Mathematicae Graph Theory, Tome 25 (2005) pp. 29-34. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1256/

[000] [1] M. Blidia, M. Chellali and T.W. Haynes, Characterizations of trees with equal paired and double domination numbers, submitted for publication. | Zbl 1100.05068

[001] [2] M. Blidia, M. Chellali, T.W. Haynes and M.A. Henning, Independent and double domination in trees, submitted for publication. | Zbl 1110.05074

[002] [3] M. Chellali and T.W. Haynes, Paired and double domination in graphs, Utilitas Math., to appear. | Zbl 1069.05058

[003] [4] J. Harant, A. Pruchnewski and M. Voigt, On dominating sets and independent sets of graphs, Combin. Prob. and Comput. 8 (1998) 547-553, doi: 10.1017/S0963548399004034. | Zbl 0959.05080

[004] [5] F. Harary and T.W. Haynes, Double domination in graphs, Ars Combin. 55 (2000) 201-213. | Zbl 0993.05104

[005] [6] T.W. Haynes, S.T. Hedetniemi and P.J. Slater, Fundamentals of Domination in Graphs (Marcel Dekker, New York, 1998). | Zbl 0890.05002

[006] [7] T.W. Haynes, S.T. Hedetniemi and P.J. Slater (eds), Domination in Graphs: Advanced Topics (Marcel Dekker, New York, 1998). | Zbl 0883.00011

[007] [8] M.A. Henning, Graphs with large double domination numbers, submitted for publication.

[008] [9] C.S. Liao and G.J. Chang, Algorithmic aspects of k-tuple domination in graphs, Taiwanese J. Math. 6 (2002) 415-420. | Zbl 1047.05032

[009] [10] C.S. Liao and G.J. Chang, k-tuple domination in graphs, Information Processing Letters 87 (2003) 45-50, doi: 10.1016/S0020-0190(03)00233-3.